Finite Group Databasebeta

← #535397

Subgroups — #535397

Order 512.

3658 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2344C211yesmaximal, characteristic, complemented
2C2×#2339C211yesmaximal, characteristic
3C2×#2311C211yesmaximal, characteristic
4#14375C214yesmaximal, complemented
5#14371C214yesmaximal, complemented
6#14142C212yesmaximal, complemented
7#14064C214yesmaximal, complemented
8#14028C214yesmaximal, complemented
9#14021C212yesmaximal, complemented
10#14018C212yesmaximal, complemented
11#13937C211yesmaximal, characteristic
12#13924C211yesmaximal, characteristic, complemented
13#13852C211yesmaximal, characteristic, complemented
14#13820C212yesmaximal, complemented
15#13777C211yesmaximal, characteristic, complemented
16C23×#41C2211yescharacteristic
17C22×C4×D4C2212yes—
18C22×#117C2212yes—
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2344—28no—
22#2339—28no—
23#2311C4116yescomplemented
24C2×#391C2212yes—
25C2×#391C2212yes—
26C2×#391C2211yescharacteristic
27C2×#391C2211yescharacteristic
28C2×#390C2212yes—
29C2×#390C2211yescharacteristic
30C2×#390C2211yescharacteristic
31C2×#389C2212yes—
32C2×#387C2214yes—
33C2×#386C2214yes—
34C2×#386C2214yes—
35C2×#385C2214yes—
36C2×#381C2212yes—
37C2×#381C2212yes—
38C2×#380C2211yescharacteristic
39C2×#380C2214yes—
40C2×#380C2211yescharacteristic
41C2×C4×C4⋊C4C2212yes—
42C2×C4×C4⋊C4C2212yes—
43C2×C4×C4⋊C4C2212yes—
44C2×C4×C22⋊C4C2211yescharacteristic
45C2×C4×C22⋊C4C2211yescharacteristic
46C2×C4×C22⋊C4C2214yes—
47C22×#95C2212yes—
48#1445C2212yes—
49#1445C2212yes—
50#1443C2214yes—
51#1443C2214yes—
52#1442C2214yes—
53#1442C2214yes—
54#1441C2212yes—
55#1441C2212yes—
56#1439C2212yes—
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1438C2212yes—
61#1438C2212yes—
62#1437C2214yes—
63#1437C2214yes—
64#1437C2214yes—
65#1436C2214yes—
66#1436C2214yes—
67#1436C2214yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2212yes—
71#1434C2214yes—
72#1434C2214yes—
73C22⋊#95C2214yes—
74C22⋊#95C2214yes—
75C22⋊#95C2214yes—
76C22⋊#95C2214yes—
77C4×#95C2212yes—
78C22×#41—28no—
79C22×#41—24no—
80C22×#41—24no—
81C22×#41—44no—
82C22×#41C2×C418yes—
83C23×Q8C2311yescharacteristic
84C23×D4C2312yes—
85C23×D4C2311yescharacteristic
86C24×C4C2312yes—
87C24×C4C2311yescharacteristic
88C24×C4C2314yes—
89C24×C4C2312yes—
90C2×C4×D4—28no—
91C2×C4×D4—28no—
92C2×C4×D4—44no—
93C2×#117—28no—
94C2×#117—44no—
95C2×#117—48no—
96C2×#117C2×C418yescomplemented
97C22×C4⋊C4—24no—
98C22×C4⋊C4—24no—
99C22×C4⋊C4C2×C418yescomplemented
100C22×C4⋊C4C2312yes—
101C22×C4⋊C4C2314yes—
102C22×C4⋊C4C2312yes—
103C22×C4⋊C4C2311yescharacteristic
104C22×C4⋊C4C2311yescharacteristic
105C22×C4⋊C4C2312yes—
106C22×C4⋊C4C2312yes—
107C22×C4⋊C4C2312yes—
108C22×C4⋊C4C2312yes—
109C22×C22⋊C4C2311yescharacteristic
110C22×C22⋊C4C2314yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2314yes—
114C22×C22⋊C4C2312yes—
115C22×C22⋊C4C2311yescharacteristic
116C22×C22⋊C4C2312yes—
117C22×C22⋊C4C2314yes—
118C22×C22⋊C4C2314yes—
119C22×C22⋊C4C2312yes—
120C22×C42C2312yes—
121C22×C42C2312yes—
122C22×C42C2314yes—
123C22×C42C2311yescharacteristic
124C22×C42C2311yescharacteristic
125C22×C42C2314yes—
126#391—28no—
127#391—44no—
128#391C2×C418yescomplemented
129#391C2×C418yescomplemented
130#390—28no—
131#390C2×C418yes—
132#390C2×C418yes—
133#389—28no—
134#387—216no—
135#386—216no—
136#386C2×C4132yescomplemented
137#385—48no—
138#381—28no—
139#381—28no—
140#380—24no—
141#380C2×C4132yescomplemented
142#380D418yes—
143C4×C4⋊C4—28no—
144C4×C4⋊C4—44no—
145C4×C4⋊C4—44no—
146C4×C22⋊C4—24no—
147C4×C22⋊C4—24no—
148C4×C22⋊C4—48no—
149C2×#95—28no—
150C2×#95C2312yes—
151C2×#95C2312yes—
152C2×#95C2312yes—
153C2×#95C2312yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2311yescharacteristic
159C2×#95C2311yescharacteristic
160C2×#95C2312yes—
161C2×#95C2312yes—
162C2×#95C2314yes—
163C2×#95C2314yes—
164C2×#95C2312yes—
165C2×#95C2314yes—
166C2×#95C2314yes—
167C2×#95C2314yes—
168C2×#95C2314yes—
169C2×#95C2314yes—
170C2×#95C2314yes—
171C2×#95C2312yes—
172C2×#95C2314yes—
173C2×#95C2314yes—
174C2×#95C2314yes—
175C2×#95C2312yes—
176C25C2411yescharacteristic
177C25C2412yes—
178C2×#41—232no—
179C2×#41—232no—
180C2×#41—416no—
181C2×#41—432no—
182C2×#41—432no—
183C22×Q8—22no—
184C22×Q8—24no—
185C22×Q8—22no—
186C22×Q8—24no—
187C22×Q8C22×C414yes—
188C22×D4—24no—
189C22×D4—28no—
190C22×D4—24no—
191C22×D4—22no—
192C22×D4—22no—
193C22×D4—24no—
194C22×D4—44no—
195C22×D4C22×C418yes—
196C22×D4C22×C414yes—
197C22×D4#4118yes—
198C23×C4—24no—
199C23×C4—24no—
200C23×C4—24no—
201C23×C4—24no—
202C23×C4—28no—
203C23×C4—24no—
204C23×C4—24no—
205C23×C4—28no—
206C23×C4—28no—
207C23×C4—28no—
208C23×C4—44no—
209C23×C4—44no—
210C23×C4—44no—
211C23×C4—44no—
212C23×C4—44no—
213C23×C4—44no—
214C23×C4—44no—
215C23×C4C22×C418yes—
216C23×C4C22×C414yes—
217C23×C4C22×C418yes—
218C23×C4C2×D414yes—
219C23×C4#4114yes—
220C23×C4C2412yes—
221C23×C4C2412yes—
222C23×C4C2414yes—
223C23×C4C2412yes—
224C23×C4C2412yes—
225C23×C4C2412yes—
226C23×C4C2412yes—
227C23×C4C2412yes—
228C23×C4C2414yes—
229C23×C4C2412yes—
230C23×C4C2412yes—
231C23×C4C2411yescharacteristic
232C23×C4C2411yescharacteristic
233C4×D4—432no—
234#117—416no—
235#117—432no—
236C2×C4⋊C4—232no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—22no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—28no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—24no—
252C2×C4⋊C4—42no—
253C2×C4⋊C4—42no—
254C2×C4⋊C4—42no—
255C2×C4⋊C4—42no—
256C2×C4⋊C4—44no—
257C2×C4⋊C4—42no—
258C2×C4⋊C4—42no—
259C2×C4⋊C4C22×C418yes—
260C2×C4⋊C4C22×C414yes—
261C2×C4⋊C4C22×C414yes—
262C2×C4⋊C4C22×C418yes—
263C2×C4⋊C4C2×D418yes—
264C2×C22⋊C4—22no—
265C2×C22⋊C4—24no—
266C2×C22⋊C4—22no—
267C2×C22⋊C4—24no—
268C2×C22⋊C4—28no—
269C2×C22⋊C4—28no—
270C2×C22⋊C4—28no—
271C2×C22⋊C4—28no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—24no—
274C2×C22⋊C4—28no—
275C2×C22⋊C4—44no—
276C2×C22⋊C4—44no—
277C2×C22⋊C4—44no—
278C2×C22⋊C4—42no—
279C2×C22⋊C4—42no—
280C2×C22⋊C4—42no—
281C2×C22⋊C4—42no—
282C2×C22⋊C4—44no—
283C2×C22⋊C4—44no—
284C2×C22⋊C4—44no—
285C2×C22⋊C4—44no—
286C2×C22⋊C4—44no—
287C2×C22⋊C4—44no—
288C2×C22⋊C4—44no—
289C2×C22⋊C4C22×C418yes—
290C2×C22⋊C4C22×C4116yes—
291C2×C22⋊C4C22×C4116yes—
292C2×C22⋊C4C2×D418yes—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—28no—
296C2×C42—24no—
297C2×C42—24no—
298C2×C42—24no—
299C2×C42—24no—
300C2×C42—28no—
301C2×C42—24no—
302C2×C42—24no—
303C2×C42—44no—
304C2×C42—44no—
305C2×C42—44no—
306C2×C42—44no—
307C2×C42C22×C414yes—
308C2×C42C22×C414yes—
309#95—22no—
310#95—28no—
311#95—22no—
312#95—24no—
313#95—28no—
314#95—28no—
315#95—28no—
316#95—24no—
317#95—24no—
318#95—24no—
319#95—44no—
320#95—42no—
321#95—42no—
322#95—44no—
323#95—44no—
324#95—44no—
325#95—42no—
326#95—42no—
327#95—44no—
328#95—44no—
329#95—44no—
330#95—44no—
331#95—44no—
332#95C22×C4116yes—
333#95C22×C4116yes—
334#95C2×D418yes—
335C24—22no—
336C24—22no—
337C24—22no—
338C24—24no—
339C24—22no—
340C24—22no—
341C24—24no—
342C24—22no—
343C24—24no—
344C24—24no—
345C24—22no—
346C24—24no—
347C24—22no—
348C24C23×C414yes—
349C24C23×C412yes—
350C24C22×D412yes—
351C24C2×#4118yes—
352C24C2×#4112yes—
353C24C2511yescharacteristic
354#41—4128no—
355C2×Q8—28no—
356C2×Q8—28no—
357C2×Q8—28no—
358C2×Q8—48no—
359C2×Q8—48no—
360C2×D4—216no—
361C2×D4—216no—
362C2×D4—28no—
363C2×D4—28no—
364C2×D4—216no—
365C2×D4—216no—
366C2×D4—48no—
367C2×D4—416no—
368C2×D4—416no—
369C2×D4#117116yescomplemented
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—28no—
373C22×C4—28no—
374C22×C4—216no—
375C22×C4—216no—
376C22×C4—216no—
377C22×C4—216no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—24no—
386C22×C4—22no—
387C22×C4—22no—
388C22×C4—24no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—24no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—22no—
413C22×C4—24no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—24no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—216no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—24no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—24no—
429C22×C4—24no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—24no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—24no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—416no—
443C22×C4—48no—
444C22×C4—416no—
445C22×C4—416no—
446C22×C4—416no—
447C22×C4—416no—
448C22×C4—416no—
449C22×C4—48no—
450C22×C4—48no—
451C22×C4—416no—
452C22×C4—88no—
453C22×C4—88no—
454C22×C4—88no—
455C22×C4—88no—
456C22×C4C4×D4116yes—
457C22×C4C23×C414yes—
458C22×C4C23×C414yes—
459C22×C4C23×C414yes—
460C22×C4C23×C414yes—
461C22×C4C23×C414yes—
462C22×C4C23×C412yes—
463C22×C4C23×C412yes—
464C22×C4C22×D418yes—
465C22×C4C22×D412yes—
466C22×C4C22×D412yes—
467C22×C4C2×#4112yes—
468C22×C4C2×#4114yes—
469C22×C4C2×#4112yes—
470C22×C4H(5,𝔽2)14yes—
471C22×C4H(5,𝔽2)14yes—
472C22×C42−1+414yes—
473C4⋊C4—216no—
474C4⋊C4—28no—
475C4⋊C4—28no—
476C4⋊C4—216no—
477C4⋊C4—48no—
478C4⋊C4—48no—
479C4⋊C4—416no—
480C4⋊C4—48no—
481C4⋊C4—48no—
482C22⋊C4—28no—
483C22⋊C4—416no—
484C22⋊C4—416no—
485C22⋊C4—48no—
486C22⋊C4—48no—
487C22⋊C4—48no—
488C22⋊C4—416no—
489C22⋊C4—416no—
490C22⋊C4—44no—
491C22⋊C4—84no—
492C22⋊C4—88no—
493C42—28no—
494C42—28no—
495C42—416no—
496C42—48no—
497C42—48no—
498C42—88no—
499C23—22no—
500C23—24no—
501C23—22no—
502C23—24no—
503C23—28no—
504C23—24no—
505C23—28no—
506C23—24no—
507C23—28no—
508C23—24no—
509C23—24no—
510C23—28no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—28no—
518C23—28no—
519C23—28no—
520C23—28no—
521C23—42no—
522C23—42no—
523C23—44no—
524C23—44no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—42no—
530C23—42no—
531C23—44no—
532C23—42no—
533C23—42no—
534C23—44no—
535C23—44no—
536C23—42no—
537C23—42no—
538C23—44no—
539C23C2×#11718yes—
540C23C2×C4×D414yes—
541C23C24×C411yescharacteristic
542C23C23×D411yescharacteristic
543C23C22×#4112yes—
544C23C22×#4111yescharacteristic
545C23C2×H(5,𝔽2)12yes—
546C23C2×H(5,𝔽2)12yes—
547C23C2×2−1+412yes—
548C23#58512yes—
549C23#58512yes—
550Q8—416no—
551D4—232no—
552D4—432no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—232no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—28no—
568C2×C4—28no—
569C2×C4—24no—
570C2×C4—24no—
571C2×C4—28no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—24no—
575C2×C4—28no—
576C2×C4—24no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—432no—
581C2×C4—432no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—42no—
587C2×C4—42no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—44no—
593C2×C4—44no—
594C2×C4—44no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—42no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—42no—
607C2×C4—42no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—42no—
612C2×C4—44no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—44no—
618C2×C4—44no—
619C2×C4—44no—
620C2×C4—42no—
621C2×C4—42no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—42no—
625C2×C4—42no—
626C2×C4—42no—
627C2×C4—44no—
628C2×C4—44no—
629C2×C4—42no—
630C2×C4—42no—
631C2×C4—832no—
632C2×C4C2×#11714yes—
633C2×C4C2×C4×D414yes—
634C2×C4C2×C4×D414yes—
635C2×C4#51814yes—
636C2×C4#54618yes—
637C2×C4#55118yes—
638C22—24no—
639C22—28no—
640C22—216no—
641C22—24no—
642C22—24no—
643C22—48no—
644C22—44no—
645C22—44no—
646C22—48no—
647C22—44no—
648C22—48no—
649C22—48no—
650C22—44no—
651C22—44no—
652C22—48no—
653C22—44no—
654C22—44no—
655C22—44no—
656C22C22×#11711yescharacteristic
657C22C22×C4×D411yescharacteristic
658C22C2×#51811yescharacteristic
659C22C4×H(5,𝔽2)12yes—
660C22C4×2−1+412yes—
661C22C2×#54612yes—
662C22C2×#55112yes—
663C22#348411yescharacteristic
664C22#348711yescharacteristic
665C22#348912yes—
666C22#350612yes—
667C22#350911yescharacteristic
668C22#351011yescharacteristic
669C22#351312yes—
670C22#351512yes—
671C22#352012yes—
672C22#352112yes—
673C22#353612yes—
674C22#353912yes—
675C22#354612yes—
676C22#354712yes—
677C4—24no—
678C4—44no—
679C4—48no—
680C4—44no—
681C4—44no—
682C4—44no—
683C4—44no—
684C4—44no—
685C4—44no—
686C4—84no—
687C4—82no—
688C4—82no—
689C4#234218yes—
690C2—44no—
691C2—48no—
692C2C2×#234211yescharacteristic
693C2#3426012yes—
694C2#3426511yescharacteristic
695C2#3426811yescharacteristic
696C2#3441012yes—
697C2#4535312yes—
698C2#4536612yes—
699C2#4536812yes—
700C2#4541212yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×#41C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#2344C2semidirect
4. #14375C2semidirect
5. #14371C2semidirect
6. #14142C2semidirect
7. #14064C2semidirect
8. #14028C2semidirect
9. #14021C2semidirect
10. #14018C2semidirect
12. #13924C2semidirect
13. #13852C2semidirect
14. #13820C2semidirect
15. #13777C2semidirect
23. #2311C4semidirect
96. C2×#117C2×C4semidirect
99. C22×C4⋊C4C2×C4semidirect
128. #391C2×C4semidirect
129. #391C2×C4semidirect
136. #386C2×C4semidirect
141. #380C2×C4semidirect
369. C2×D4#117semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23442
C2×#23392
C2×#23112
#143752
#143712
#141422
#140642
#140282
#140212
#140182
#139372
#139242
#138522
#138202
#137772